Motivation

“The study of form may be descriptive merely, or it may become analytical. We begin by describing the shape of an object in simple words of common speech: we end by defining it in the precise language of mathematics; and the one method tends to follow the other in strict scientific order and historical continuity.”

D’Arcy Thompson. (1915). Trans. Roy. Soc. Edinburgh

Morphometrics

• What is morphometrics?
• From Greek: morpho (form) + metrikos (measure)
• The quantification and analysis of morphological objects
• Application of multivariate statistical analysis to the study of variation in plants and animals (Reyment, 1985)

• The study of size and shape

• The quantification of form (Lestrel, 2000)

• The study of shape variation and its covariation with other variables (Bookstein, 1991)

• This workshop is concerned with the quantification of shape as derived from landmark-based methods, and a proper evaluation of patterns of shape variation relative to explanatory variables

Why Study Shape?

• Historical reasons: natural history studies are largely morphological
• Shape differences delineate obvious biological groups (i.e. alpha-taxonomy)
• Shape covaries with other variables
• Understanding shape and biological form may lead to understanding function

Why Study Shape?

• Historical reasons: natural history studies are largely morphological
• Shape differences delineate obvious biological groups (i.e. alpha-taxonomy)
• Shape covaries with other variables
• Understanding shape and biological form may lead to understanding function

To properly study shape we require a definition of it

• The geometric properties of an object that are invariant to changes in translation, rotation, and scale (Bookstein 1991); Dryden and Mardia (1998)

• or… The variation that remains once position, orientation, and scale have been mathematically held constant

Form and Shape

• For much of the history of biological inquiry, form and shape were synonymous

• Plato & Aristotle: believed that understanding form lead to understanding of function

• Geothe: ‘morphology’ the form of an organism

• Form and shape historically used interchangeably

• Now, more formal definitions of form and shape exist, which provide important distinctions

• Specifically, Form = size + shape (current definition)

Early Uses of Morphology

• For centuries, biologists used morphology to compare organisms

• Species taxonomy, classification, and ecological specialization were all addressed using anatomical properties

• However, morphology was typically described qualitatively, leading to limitations

• Much of the richness of morphological variation not captured

• Limits comparisons to 2 objects (same or different) For > 2 objects, only rank-orderings could be done (A >B >C)

• Rank-ordering implies differences on a continuous scale

• Thus, there was a need to quantitatively compare morphology

• This moves towards a notion of shape and shape differences

Transformation Grids in Art: Durer (1528)

• An intuitive use of shape differences from art!

Transformation Grids in Biology

• Early biological ideas

Early versions were qualitative. This week we learn the thin-plate spline which enables accurate shape transformations

The Biometric Tradition

• Use of quantitative data began in the late 1800s: variables were measured (e.g., wing length) and group means compared

• Morphometric advances concurrent with statistical advances

• LATE 1800s – EARLY 1900s

• Galton: variation, correlation, regression

• Pearson: more on correlation, \(\small\chi^2\)

• From this the Biometric tradition was formed, believing that most of biological variation was continuous

• Early 1900s: BIG biological debate: Biometric vs. Mendelian (continuous vs. discontinuous biological variation)

Development of Statistics

• Statistics developed (as a field) with evolutionary biology

• ANOVA / Variance partitioning (Fisher: for quantitative genetics and more generally)

• Discriminant Function Analysis (DFA/CVA), MANOVA,

• Generalized distance (Fisher, Mahalanobis, Rao)

• Principal Components Analysis (Pearson, Hotelling)

• Multivariate t-test (Hotelling)

• Factor Analysis (Spearman)

• Path Analysis (Wright)

• Most data used were biological

• THIS paved the way for morphometrics

Multivariate Morphometrics

• Multivariate analysis of morphological measurements

• Within- and between-group variation described and compared

• Commonly used methods: principal components, canonical variates, discriminant function (see Blackith and Reyment, 1971)

Multivariate Morphometrics

• Multivariate analysis of morphological measurements

• Within- and between-group variation described and compared

• Commonly used methods: principal components, canonical variates, discriminant function (see Blackith and Reyment, 1971)

• Related topics:
  • Numerical Taxonomy: Sokal and Sneath, 1963; Sneath and Sokal, 1973
  • Size, shape and allometry (size vs. shape): Jolicoeur, 1963; Burnaby, 1966; Mosimann, 1970

Geometric Morphometrics

• 1980s-1990s: Theoretical advances and data types caused radical shift in morphometric methods

• Geometric Morphometric Methods (GMM) emerged

• Rigorous quantification of shape coupled with sound statistics and graphical visualization

• Emphasis on landmarks due to more formal statistical theory

• Procrustes Paradigm: rigorous quantification of shape from points, curves, and surfaces

• Analysis of shape variation with graphical description of results

Measuring Shapes

• Morphometrics: the study of shape variation and its covariation

• How to quantify shape? There are no natural units for shape

Measuring Shapes

• Morphometrics: the study of shape variation and its covariation

• How to quantify shape? There are no natural units for shape
  • Use a set of proxy variables to represent shape (morphometric traits)
  • Different approaches use different types of variables and methodologies to obtain them

Measuring Shapes

• Morphometrics: the study of shape variation and its covariation

• How to quantify shape? There are no natural units for shape
  • Use a set of proxy variables to represent shape (morphometric traits)
  • Different approaches use different types of variables and methodologies to obtain them

• Data should

• Register shape in a repeatable manner and archive it for statistical analyses

• Contain enough information to reconstruct a graphical representation of the structure of interest

• Be appropriate to address the biological question of interest

Morphometric Data

• Morphometric data may quantify:

• The location of discrete anatomic features (points)

• Distances between anatomic features (lines)

• Outlines of a structure (curves)

• The surface of a structure (surfaces)

• The modern morphometric synthesis permits the combination of ALL these types of data

Homology in Morphometrics

• Data must quantify shape in a repeatable manner

• Homology: correspondence across specimens

• Evolutionary homology: structures derived from the same tissue of the most recent common ancestor

• Operational homology: correspondence in the position of structures

• In Morphometrics, we may use both, depending on the question to answer

• The important point is to obtain repeatable data

Data Types: Linear Measurements

• Use linear measurements, angles, etc., between anatomic features, or structures (e.g. head width)

• Use some method to eliminate non-shape variation (size)

• Analyze the data using multivariate statistics

• aka ‘traditional’ or “multivariate” morphometrics

Linear Measurements: Considerations

Advantages

• Allow comparison to previous studies

• The variables used are very intuitive and easy to interpret biologically

Disadvantages

• Size is a latent factor and there is no global consensus on how to account for it – different approaches provide different results

• The same values may represent different shapes

• Homology is difficult to evaluate and guarantee

• Usually it is difficult to obtain a graphical representation of variation in shape (because the geometry of the structure is not preserved in the analysis)

Linear Measures: Same Values Different Shapes

Linear Measures: Same Values Different Shapes

Something is missing! The relative positions of the distances on the structure

The Truss

• An interconnected network of distance measurements between anatomical points

• An effort to maintain the relative positions between traits

• Can provide a graphical representation

• BUT statistical treatment is problematic

The Morphometric Revolution

• In the 1980s, there is a radical shift in methodology

• Linear measurements do not capture all of the properties required to describe shape

• The truss was an attempt to record relative positions

• This information is inherent in the common endpoints of linear distances

• Change from linear measurements to the locations of points themselves

• Use landmark coordinates as the raw data

This advance was aided by the simultaneous development of mathematical shape theory

Data Types: Landmark Coordinates

• Use x, y (or x, y, z) coordinates of anatomic locations to quantify anatomical variation

• Eliminate non-shape variation (size and others, see further on)

• Multivariate data analysis

• ‘Landmarks are the points at which one’s explanations of biological processes are grounded.’ Bookstein (1990)

Types of Landmarks

Landmarks: Considerations

Advantages

• Easy to assess homology, so biological interpretation is more robust

• The geometry of the studied structure is maintained in relative landmark locations

• Can obtain a visual representation of shape variation

Disadvantages

• Raw data include information that is NOT about form (e.g. location, scale, orientation)
• Some structures have few or no landmarks

These issues are alleviated through Geometric Morphometric methods and its extensions to other data types, see further on

Data Types: Curves

• Some structures have few, or no, landmarks

• One may be interested in outlines, or other curves, to quantify shape

• Types of curves
  • Partial: different starting and ending points
  • Complete: the starting and ending point coincide (closed curves)

Curves: Considerations

Advantages

• Useful when there are few, or no, landmarks

• They register a full representation of the outline (and therefore we can obtain a graphical representation of shape variation)

Disadvantages

• Difficult to assess homology

• Several analytical approaches exist, which provide different variables, and comparison among them is not straightforward (there is no globally accepted way to measure the shape of a curve)

• Most methods do not allow the combination of curves and landmarks

• Some curves may represent different shapes

But see next! – The Procrustes paradigm allows the combination of points, curves and surfaces

Semilandmarks

• They resolve the problems encountered when working with curves

• Can be quantified on curves (2D) or surfaces (3D)

• They are treated as “degenerate” landmarks, the position of which is limited to slide on a curve (or surface) during analysis

• Only variation perpendicular to the curve is relevant

Semilandmarks: Considerations

Advantages

• Good representation of curves, outlines and surfaces

• They allow the examination of curves and surfaces in Procrustes shape space (which has known statistical properties)

• It is possible to combine different morphological features (points, curves, surfaces) in a single analysis of form

The Procrustes Paradigm

• Combining landmarks and semilandmarks there is a general solution for quantifying biological shape variation

• We obtain quantitative information that represents:
  • Discrete anatomical points
  • Curves (outlines) of structures
  • Surfaces of structures

• We can combine it all in a single analysis!

General Morphometric Protocol

Special Considerations: Missing Data

• Sometimes landmarks are missing

• Methods to estimate their location exist (discussed later this week)

Special Considerations: Symmetry

• Sometimes objects display symmetry

• GMM procedures adjusted slightly for this (discussed later this week)

Dean’s View of Quantitative Biology

• Methods are a product of theory and data type

• Choose methods that are mathematically sound, but also theoretically and biologically sound

• METHODS SHOULD NOT INTRODUCE PATTERNS!!!!